Wednesday, January 2, 2008

Wednesday Math, Vol. 9: phi, the golden ratio

A rectangle is often defined by two numbers, the height and width. For example, an index card is 3"x 5", a standard piece of paper is 8.5" x 11", a common thickness for lumber used in construction is 2" x 4". For mathematicians, we sometimes simplify this even further to a single number, the ratio big/little. In the examples above, the 3" x 5" would be a 5/3 = 1.66... rectangle, the 8.5" x 11" would be a 11/8.5 = 1.2941... rectangle and the 2" x 4" would be a 4/2 = 2 rectangle.

If two rectangles have the same ratio of big to little, they are called similar. None of these examples is similar to any of the others. A 6 x 10 rectangle would be similar to a 3 x 5, because 5/3 = 10/6.

If we have a rectangle that isn't a square, we can always split it into a square and smaller rectangle. For example, the 3" x 5" could be split into a 3" square with a 2" x 3" leftover rectangle. In this case, the 3" x 5" is a 5/3 = 1.66... ratio and the 2" x 3" is a 3/2 = 1.5 ratio. The big rectangle and the small rectangle are not similar. There is only one ratio where the big rectangle and the small rectangle will have the ratio, and that ratio is known by the Greek letter phi (pronounced "fee") and the exact value is (1+sqrt(5))/2 = 1.6180339887..., a non-repeating decimal because it is irrational.

The Greeks thought this number was really cool and used it a lot in their artwork, in the form of the golden rectangle.

The Renaissance was in many ways the rediscovery of the Greek work done before the birth of Christ. Artists decided to emulate the work of the ancient Greeks, including using the golden ratio. Leonardo da Vinci was particularly keen on it.

With the golden rectangle, we can remove a square and be left with a new smaller golden rectangle ad infinitum. If we line up the squares correctly and put a quarter circle in each of the squares in the proper orientation, we get a shape known as the Archimedan Spiral, named for the total badass mathematician Archimedes.

This spiral shows up in nature in the shape of the nautilus shell.

If we draw a regular pentagon, every side the same length and every angle 108 degrees, then the length of the diagonal (in this case the line segment from point B to point E) has a golden ratio relationship with the common length of the sides.

Because phi is an irrational number, it becomes hard to represent exactly in a computer. If I want to draw a golden rectangle on a piece of paper, there is a construction method using a straightedge and a compass, but if I want to use a computer, the pixels (short for picture elements) have to be at an exact whole number of pixels apart. This is where the Fibonacci numbers come in. Let's look at one Fibonacci number divided by the previous Fibonacci number and compare these to phi.

None of these ratios is exactly phi, and none of them ever can be, but they get closer and closer and they alternate eternally between being too high and too low. A rectangle with sides that are two consecutive Fibonacci numbers is as close as you can get to golden rectangle using whole numbers, like the 55 by 34 rectangle I drew above.

And now I will let you in on a little Matty Boy secret. If the picture allows it, I often crop the photos in my blog to be as close to golden rectangles as possible.

well-represented, Matty-boy. i too like the 2:3 aspect ratio because it works with phi, or as close as one can get to phi in pixels, anyway. i don't crop the news photos i use on my blog, but my own artwork tends to 2:3 and phi.

i enjoyed this post very much. it was a nice read, compared to the other stuff i'm doing today. thanks!

Yes, Dan Brown has ruined the Fibonacci sequence for many. It's kind of like making fun of the French. At one time, it was a wellspring of American humor, but now it's been overrun by the Freedom Fries crowd.

sigh.

That said, I have a post tomorrow which will make fun of the French in a progressive and non-jingoistic way. Stay tuned.